Actuator Saturation in Individual Blade Control of Rotorcraftdsbaero/library/Conference...Actuator...

Actuator Saturation in Individual Blade Control of

Rotorcraft

Ashwani K. Padthe�, Peretz P. Friedmanny, and Dennis S. Bernsteinz

Department of Aerospace Engineering

The University of Michigan, Ann Arbor, MI, 48109

The e�ect of actuator saturation on the vibration and noise reduction capabilities ofactively controlled trailing-edge aps and micro aps is examined. A new approach tohandling actuator saturation in the HHC algorithm is developed based on constrainednonlinear optimization techniques. The performance of this approach in reducing vibra-tions and noise is compared to three existing approaches at various ight conditions. Theresults indicate that truncating or scaling the optimal ap/micro ap de ection can sig-ni�cantly compromise the vibration or noise reduction performance. By comparison, theauto-weighting approach and the new optimization based approach yield far better perfor-mance. However, the optimization approach takes less computational time and performsbetter in the case of multiple control surfaces as it utilizes all of them to the maximumpossible extent.

Nomenclature

b Airfoil semi-chord = c=2c Airfoil chordcp Pressure coe�cient on the airfoil surfaceCT Rotor thrust coe�cientCdf Parasitic drag coe�cient of the fuselageCW Helicopter Weight coe�cientD0; D1 Generalized ap motionse Blade root o�set from the center of rotationf Generalized load vectorFHX4; FHY 4;

FHZ4 Nondimensional 4/rev hub shearsk Reduced frequency = !b=ULb Blade lengthLc Spanwise dimension of blade segment with micro apM Mach numberMb Blade massMHX4;MHY 4;

MHZ4 Nondimensional 4/rev hub momentsMHz1 Total yawing moment about the hubNb Number of rotor bladesNH06; NH07;� � � ; NH17 6th � 17th harmonic components of BVI noisenL Number of lag termsPR Rotor power�p Nondim. surface pressure distribution

�Postdoctoral Researcher, Member AIAA.yFran�cois-Xavier Bagnoud Professor, Fellow AIAA.zProfessor, Member AIAA.

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American Institute of Aeronautics and Astronautics

53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR>20th AI23 - 26 April 2012, Honolulu, Hawaii

AIAA 2012-1477

Copyright © 2012 by A. K. Padthe, P. P. Friedmann and D. S. Bernstein. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Q Output weightingR Control input weightingR Blade radiuss Laplace variable�s Nondim. Laplace variable = sb=UT Sensitivity matrixt Time�t Reduced time = 1

b

R t0U(�)d�

U(t) Freestream velocity, time-dependentu Control input vectorv;�v;�va Components of the local velocity at any point on the surface of the airfoil due to thickness,

camber, and angle of attack, respectivelyV Freestream velocity for the airfoilw Disturbancew Relation between the output and the disturbanceW� Relative weighting parameterXA O�set between the aerodynamic center and the elastic axisXFA; ZFA Longitudinal and vertical o�sets between rotor hub and helicopter aerodynamic centerXFC ; ZFC Longitudinal and vertical o�sets between rotor hub and helicopter center of gravityXIb O�set of the blade cross-sectional center of mass from the elastic axisxc Spanwise location of center of micro ap segmentz Output vector� Airfoil angle of attack�R Rotor shaft angle�p Blade precone angle� Flap de ection�NC ; �NS N/rev cosine and sine components of ap de ection�limit Saturation limit on ap de ection Lock number n Rational approximant poles� Helicopter advance ratio! Oscillation frequency�! Nondim. normal velocity distribution Rotor angular speed!F ; !L; !T Blade ap, lead-lag and torsional natural frequencies Azimuth angle� Rotor solidity�tw Built-in twist angle

I. Introduction

Noise and vibrations are a major source of concern in the design and maintenance of helicopters. Vibra-tions cause crew and passenger discomfort and reduce the airframe and component fatigue lives resultingin high maintenance costs. High levels of blade-vortex interaction (BVI) noise are a primary hindranceto the community acceptance of a civilian helicopter. During the last two decades, various active controlapproaches, such as the higher harmonic control,1 individual blade control (IBC),2,3 actively controlled con-ventional plain trailing-edge aps (ACF),4{7 the active twist rotor (ATR),8,9 and the micro aps10{12 havebeen explored and shown to have potential for vibration and noise reduction in rotorcraft. Approaches suchas the IBC and the ATR which rely on de ecting or twisting the entire blade require high actuation power.By contrast, the ACFs (depicted in Fig. 1) and the micro aps (depicted in Fig. 2) require signi�cantly lesspower for actuation. However, the actuation devices used to operate the ACFs have limited torque capacitiesand hence are subject to amplitude saturation. Similarly for micro aps, the thickness of the airfoil imposes alimitation on the size of the micro ap thus constraining the maximum de ection of the micro ap. Saturationconstraints introduce nonlinearities into a linear system, forcing the control-system to operate in a modefor which it was not designed resulting in a signi�cant degradation of its performance.13 Therefore, it is

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important to consider the e�ects of actuator saturation in the various active control approaches.

δfα

20%c

Figure 1. A 20%c conventional plain ap cross-section.

1.5%c

0.6%c0.3%c

6%c

δf

Figure 2. An oscillating micro ap cross-section.

The higher harmonic control (HHC) algorithm has been widely employed for closed-loop vibration andnoise reduction studies using the conventional trailing-edge aps4,6, 14,15 and more recently the micro aps.11

These studies have shown that the vibration and noise reduction performance of the aps (or micro aps) isbest when implemented in a dual ap con�guration. A good review of the HHC algorithm and its variants waswritten by Johnson in Ref. 16. A detailed description of the algorithm, including robustness and stabilityanalyses, can be found in Ref. 15. However, the e�ect of actuator saturation on the performance of theHHC algorithm has received only limited attention. A preliminary investigation of the e�ect of actuatorsaturation on the vibration reduction capabilities of the HHC algorithm using a conventional plain apwas considered in Ref. 17. Three di�erent approaches for constraining the ap de ections involving ana posteriori modi�cation of the optimal control input obtained from the HHC algorithm were examined.These approaches were denoted as 1) truncation (TR) - simply clipping the optimal ap de ection wheneverit exceeds the saturation limits, 2) scaling (SC) - uniformly scaling down the optimal ap de ection such thatit never exceeds the saturation limits, and 3) auto-weighting (AW) - consisting of an iterative adjustment ofthe control weighting matrix in the HHC algorithm such that ap de ection is properly constrained. TheTR and SC approaches were inconsistent and produced only limited vibration reduction whereas the AWapproach produced excellent performance resulting in a 90% reduction of vibration. Despite its e�ectiveness,the AW approach has shortcomings. It uses an a priori guess of the upper bound on the optimal controlweighting, and requires several iterations to converge, thus increasing the computational cost. Furthermore,for multiple aps (or micro aps), the control weighting on the multiple control surfaces is identical. Thuscontrol input is optimal only for one of the control surfaces leaving the others under-utilized. Using a di�erentcontrol weighting for each of the di�erent control surfaces implies signi�cant increases in computational cost.

The shortcomings of the AW approach can be remedied by a new approach based on constrained non-linear optimization developed in this paper. In this approach, actuator saturation constraints formulated asinequality constraints on the control surface de ection are combined with the minimization of the quadraticcost function in the HHC algorithm resulting in a constrained nonlinear optimization problem. Unlike theapproaches described in Ref. 17, this approach, denoted here as the optimization (OPT) approach, accountsfor the presence of saturation nonlinearities in an a priori manner involving direct modi�cations to theHHC algorithm. A similar approach to handling actuator saturation was proposed in Ref. 18 for vibrationreduction using a single trailing-edge ap. However, the performance of this approach that is particularlye�ective for con�gurations involving multiple control surfaces was not considered.

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The overall objective of this paper is to examine the OPT approach in detail and compare it to the TR,SC, and AW approaches. The speci�c objectives are:

1. Develop a computationally e�cient approach to handling actuator saturation in the HHC algorithm,that is suitable for multiple control surfaces, without compromising on the controller performance.

2. Compare the computational cost and performance of the various actuator saturation approaches forreducing BVI noise and vibrations on a representative rotor con�guration using the single and dualcon�gurations of a 20%c conventional plain trailing-edge ap and a 1.5%c micro ap.

3. Compare the vibration reduction performance of the various saturation approaches at two di�erent ight conditions: (a) low-speed descending ight where vibrations are high due to blade-vortex inter-action and (b) high-speed cruise condition.

II. Description of the Aeroelastic Analysis Code

The comprehensive rotorcraft simulation code AVINOR (Active Vibration and Noise Reduction), whichhas been extensively validated in earlier studies,6,11,19,20 is used for all the active vibration reduction studiespresented in this paper. A brief description of the various components of the code is provided next, foradditional details see Ref. 19.

II.A. Aerodynamic model

The blade/ ap and blade/micro ap sectional aerodynamic loads for attached ow are calculated using a ra-tional function approximation (RFA) based nonlinear reduced order model constructed from CFD data.10,21

This model provides unsteady lift, moment, as well as drag predictions for both plain ap and micro apcon�gurations. The CFD based RFA model is linked to a free wake model,6,20 which produces a spanwiseand azimuthally varying in ow distribution. For separated ow regime, the aerodynamic loads are calculatedusing the ONERA dynamic stall model.6

II.B. Structural dynamic model

The structural dynamic model consists of a four-bladed hingeless rotor, having fully coupled ap-lag-torsionaldynamics for each blade. The structural dynamic model is geometrically nonlinear, due to moderate bladede ections. The structural equations of motion are discretized using the global Galerkin method, basedupon the free vibration modes of the rotating blade. The dynamics of the blade are represented by three ap, two lead-lag, and two torsional modes. Free vibration modes of the blade were obtained using the �rstnine exact non-rotating modes of a uniform cantilevered beam. The e�ects of control surfaces such as thetrailing-edge plain ap or the micro ap on the structural properties of the blade were neglected. Thus, thecontrol surfaces in uence the blade behavior only through their e�ect on the aerodynamic and inertial loads.

II.C. Coupled aeroelastic response/trim solution

The combined structural and aerodynamic equations are represented by a system of coupled ordinary dif-ferential equations with periodic coe�cients in state-variable form. The trim employed is a propulsive trimprocedure where three force equations (longitudinal, lateral, and vertical) and three moment equations (roll,pitch, and yaw) corresponding to a helicopter in free ight are enforced. A simpli�ed tail rotor model, basedon uniform in ow and blade element theory, is used. The six trim variables are the rotor shaft angle �R,the collective pitch �0, the cyclic pitch �1s and �1c, the tail rotor constant pitch �0t, and lateral roll angle�R. The coupled trim/aeroelastic equations are solved in time using the ODE solver DDEABM, which is apredictor-corrector based Adams-Bashforth di�erential system solver.

II.D. Acoustic model

The acoustic calculations are based on a modi�ed version of the WOPWOP code, where helicopter noise isobtained from the Ffowcs-Williams Hawkings equation with the quadrupole term neglected.22 The version ofWOPWOP used in the code was modi�ed to account for a fully exible blade model that is compatible withthe structural dynamic model described earlier. In previous studies,6,23 the chordwise pressure distribution

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on the surface of the blade, a required input to the acoustic computations, was obtained using an extendedRFA approach. The extended RFA approach used frequency domain pressure data obtained from the doubletlattice ow solver, described in detail in Ref. 23. Generating the extended RFA models using CFD basedpressure distribution data is computationally expensive. To reduce the cost the blade pressure distributionsare obtained from an approximate velocity superposition method.24 Based on potential ow theory, thepressure distribution on the surface of the airfoil is related to the local velocity distribution that is assumedto result from three independent contributions:

Cp =

�v

V� �v

V� �va

V

�2

; (1)

where the velocity ratiosv

V,

�v

V, and

�vaV

are contributions due to airfoil thickness, camber, and angle of

attack, respectively. The signs in Eq. 1 are positive for the upper surface and negative for the lower surface

of the airfoil. For the symmetric NACA 0012 airfoil used in the study,�v

V= 0, and the values of the other

two components are found from the approach described in Ref. 24. Since this approach is based on thepotential ow theory it is not quite compatible with the CFD based RFA model. However, it represents anacceptable approximation because the lift coe�cients from which the pressure distributions are obtained,are based on the CFD+RFA model that accounts for compressibility, viscosity, and unsteady e�ects.

III. The Higher Harmonic Control Algorithm

Active control of vibration and noise is implemented using the HHC algorithm used extensively foractive control of vibration and noise in rotorcraft.6,15 The algorithm is based on the assumption that thehelicopter can be represented by a linear model relating the output of interest z to the control input u.The measurement of the plant output and update of the control input are not performed continuously, butrather at speci�c times tk = k� , where � is the time interval between updates during which the plant outputreaches a steady state. In actual implementation of the algorithm, this time interval may be one or morerevolutions. A schematic of the HHC architecture as implemented on a helicopter is shown in Fig. 3. The

Figure 3. Higher harmonic control architecture

disturbance w is representative of the helicopter operating condition. The output vector at the kth timestep is given by

zk = Tuk + Ww (2)

where the sensitivity T is given by

T =@z

@u: (3)

At the initial condition, k = 0,z0 = Tu0 + Ww: (4)

Subtracting Eq. (4) from Eq. (2) to eliminate the unknown w yields

zk = z0 + T(uk � u0): (5)

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The controller is based on the minimization of a general quadratic cost function

J(zk;uk) = zTkQzk + 2zTk Suk + uTkRuk: (6)

However, in most applications, the cross-weighting term in Eq. (6) is neglected and the cost function simpli�esto

J(zk;uk) = zTkQzk + uTkRuk: (7)

The optimal control input is determined from the requirement

@J(zk;uk)

@uk= 0; (8)

which yields the optimal control law uk;opt, given by

uk;opt = �(TTQT + R)�1(TTQ)(z0 �Tu0): (9)

Combining Eqs. (5), (7) and (9), the minimum cost is found to be

J(zk;uk;opt) = (z0 �Tu0)T�Q� (QT)D�1(TTQ)

�(z0 �Tu0): (10)

whereD = TTQT + R (11)

This is a classical version of the hhc algorithm that yields an explicit relation for the optimal control input.Another version of the HHC algorithm where the sensitivity matrix T is updated using least-squares methodsafter every control update is known as the adaptive or recursive HHC and is discussed in Ref. 15.

For a 4-bladed rotor, the control input uk is a combination of 2/rev, 3/rev, 4/rev, and 5/rev harmonicamplitudes of the control surface de ection:

uk = [�2c; �2s; :::; �5c; �5s]T : (12)

The term ‘control surface’ refers to both the micro ap and the conventional plain trailing-edge ap. Thetotal control surface de ection is given by

�( ;uk) =

5XN=2

[�Nc cos(N ) + �Ns sin(N )] : (13)

where the quantities �Nc and �Ns correspond to the cosine and sine components of the N/rev control inputharmonic. When multiple control surfaces are used, the control surface de ections are given by

�i( ;uk) =

5XN=2

[�Nci cos(N ) + �Nsi sin(N )] ; (14)

(15)

where i = 1; : : : ; N� and N� is the total number of control surfaces. The control vector uk is then given by

uk = [�2c1; �2s1; :::; �5c1; �5s1; : : : ; �2cN� ; �2sN� ; :::; �5cN� ; �5sN� ]T : (16)

For vibration reduction (VR) studies, the output vector zk consists of 4/rev vibratory hub shears andmoments:

zvr =

2666666664

FHX4

FHY 4

FHZ4

MHX4

MHY 4

MHZ4

3777777775(17)

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The weighting matrix Q in the cost function (see Eq. 7) is a diagonal matrix, and, for vibration control,is described by six weights corresponding to the three vibratory hub shears and the three vibratory hubmoments. Based on previous studies,6,11 the weights for the hub shears were assumed to be identical, and asimilar assumption was used for the weights of the hub moments. The weighting matrix used in this studyfor vibration reduction, Qvr, is:

Qvr =

2666666664

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 10 0 0

0 0 0 0 10 0

0 0 0 0 0 10

3777777775: (18)

For BVI noise reduction (NR) studies, the output vector consists of the 6th-17th blade passage frequencyharmonic components of BVI noise (the most signi�cant part of BVI noise), as measured by a microphoneinstalled at a suitable location. This location is usually on the skid or landing gear of the helicopter, and

znr =

26666664NH06

NH07

NH08

...

NH17

37777775 (19)

The noise control law is identical to the control law used for vibration reduction. The weighting matrix usedin this study for NR is:

Qnr =

2666666666664

1 0 0 0 : : : : : : 0

0 1 0 0 : : : : : : 0

0 0 1 0 : : : : : : 0. . .

0 : : : : : : 0 1 0 0

0 : : : : : : 0 0 1 0

0 : : : : : : 0 0 0 1

3777777777775: (20)

Note that all the components of the BVI noise are weighted equally.For simultaneous vibration and noise reduction (SR) problems, a combined output vector is de�ned by

zsr =

"zvr

znr

#; (21)

where the vector zsr is simply a partitioned combination of the vibration and noise levels. The combinedweighting matrix Qsr is de�ned as

Qsr =

"(W�) � [Qvr] 0

0 (1�W�) � [Qnr]

#; (22)

where W� is a scalar factor used to adjust the relative weighting between noise and vibration as objectivesfor the controller. When W� = 1, the control e�ort is focused on vibration reduction, and when W� = 0,only noise is reduced by the controller. During approach to landing, BVI noise is the main priority, whilevibration is the goal at cruise conditions. Therefore, in an actual implementation the weighting factor canbe adjusted by the controller depending on the desired outcome. The weighting matrices Qvr and Qnr used

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for simultaneous noise and vibration reduction performed in this study are:

Qvr =

2666666664

1 0 0 0 0 0

0 1 0 0 0 0

0 0 10 0 0 0

0 0 0 10 0 0

0 0 0 0 10 0

0 0 0 0 0 10

3777777775; (23)

and

Qnr =

2666666666664

100 0 0 0 : : : : : : 0

0 100 0 0 : : : : : : 0

0 0 100 0 : : : : : : 0. . .

0 : : : : : : 0 100 0 0

0 : : : : : : 0 0 100 0

0 : : : : : : 0 0 0 100

3777777777775: (24)

IV. Actuator Saturation

Most actuation devices used for on blade control of rotorcraft vibrations and noise are subject to amplitudesaturation. Furthermore the actuation amplitudes have to be limited so as to avoid undesirable interactionsbetween the primary ight control system and the on blade controller. For a micro ap the maximumde ection is constrained by its size, usually 1:5% of the chord. For a conventional trailing-edge ap themaximum de ection is set to 4�. To study the e�ects of saturation on the controller performance, fourdi�erent approaches to implementing actuator saturation in the HHC algorithm are considered: these aretruncation, scaling, auto-weighting, and optimization. In the truncation approach, the unconstrained optimalcontrol input is clipped whenever it exceeds the limiting amplitude, thus the control surface de ection is

�( ;uk) =

(�( ;uk); j�( ;uk)j < �limit

sgn(�( ;uk)) � �limit; j�( ;uk)j � �limit

(25)

where �limit is the saturation limit on the control surface de ection.In the scaling approach, the control input is given by

�( ;uk) =�limit

max(j�opt( ;uk)j)� �opt( ;uk); (26)

where �opt( ;uk) is the optimal control input obtained using the HHC algorithm without the saturationconstraints. Each harmonic component of the optimal control surface de ection is scaled by a commonfactor such that the maximum de ection is equal to the saturation limit.

In the third approach denoted as auto-weighting the control weighting matrix, R in Eq. (7), is updated soas to restrict the control surface de ection. The control weighting matrix R penalizes the control input andthus can be used to constrain the maximum control surface de ection. However, the value of R required toconstrain the control input amplitude within the saturation limits is not known a priori. Hence, an iterativeapproach which adjusts the value of R is used. The weighting matrix R is represented as:

R = cwuI: (27)

where cwu is a scalar and I is the identity matrix. In this approach all harmonic components of the controlinput vector are weighted equally. If the control surface de ection is overconstrained, the controller reducesthe value of cwu. If the control surface de ection is underconstrained, the controller increases the value ofcwu. A new optimal control is calculated using the updated value of cwu, obtained as follows:

1. Set c�wu = 0 and c+wu = cmax.

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2. Set cwu = 12 (c�wu + c+wu)

3. Calculate a new optimal control input.

If the ap de ection is properly constrained (j�maxj = �limit � 5%), end the algorithm.

If the ap de ection is underconstrained (j�maxj > �limit), set c�wu = 12 (c�wu + c+wu). Go to step 2.

If the ap de ection is overconstrained (j�maxj < �limit), set c+wu = 12 (c�wu + c+wu). Go to step 2.

This iterative procedure increases or decreases cwu until the optimal control surface de ection convergesto the desired de ection limits within a prescribed tolerance. The value of cmax has to be estimated initiallyand it has to be greater than or equal to the optimum value of cwu that properly constrains the controlinput. Choosing a very large value for cmax is not recommended since depending on the proximity of cmax

to the optimum cwu, the AW approach can take several iterations causing an increase in the computationalcosts. Furthermore, for the case of multiple control surfaces, the number of iterations required for all ofthem to be properly constrained can be quite high rendering the AW approach impractical. To avoid thissituation the same value of cwu is used for all the control surfaces.

A new approach for dealing with actuator saturation in the HHC algorithm is developed in this study.This approach, based on constrained nonlinear optimization techniques, overcomes the limitations associatedwith the previous approaches. Recall that the HHC algorithm is based on the minimization of a quadraticcost function, given by Eq. (7). The saturation limits can be combined with the minimization of the costfunction to yield a constrained optimization problem:

minimizeuk

J(zk;uk) = zTkQzk + uTkRuk; (28)

subject to j�i( ;uk)j � �limit; i = 1; : : : ; N� (29)

where N� is the total number of control surfaces. The optimization problem given by Eqs. (28) and (29) isa nonlinear constrained optimization problem with a quadratic objective function and nonlinear inequalityconstraints, and represents a Nonlinear Programming (NP) problem. Unlike the approaches described earlier,this saturation approach involves direct modi�cations to the HHC algorithm to account for the presence ofsaturation in an a priori manner. The resulting optimal control input always satis�es the saturation limitsirrespective of the values of R and Q.

A NP method, Sequential Quadratic Programming (SQP),25,26 available in the FMINCON tool in MAT-LAB, is used to solve the optimization problem given by Eqs. (28) and (29). The SQP method solves aquadratic programming subproblem based on a quadratic approximation of the Lagrangian function. Astand-alone application (a .exe �le) capable of performing the optimization is generated using the mcc -mcommand in Matlab. Subsequently, this application is invoked from the AVINOR code, written in FOR-TRAN, in order to evaluate the optimum uk. The stand-alone application requires approximately 1 sec torun on a 2.53 GHz Intel Xeon processor in the case of a single control surface. Note that the nonlinearconstraints described in Eq. (29) have to be satis�ed for all values of the azimuth angle 2 [0� 360�]. Inactual numerical implementation, the nonlinear constraints are evaluated and enforced at every integer valueof over the range [0� 360�].

V. Results and Discussion

The results presented in this section are obtained for a helicopter con�guration resembling a four-bladedMBB BO-105 hingeless rotor. The rotor parameters are listed in Table 1. All the values in the table (exceptCW ; ; and �) have been non-dimensionalized using Mb; Lb, and 1= for mass, length and time, respectively.The mass and sti�ness distributions are assumed to be constant along the span of the blade.

The vibratory hub shears and moments are obtained from the integration of the distributed inertial andaerodynamic loads over the entire blade span in the rotating frame. Subsequently, the loads are transformedto the hub-�xed non-rotating system, and the contributions from the individual blades are combined. In thisprocess, the blades are assumed to be identical. Reduction is performed on the Nb/rev components, whichare the dominant components, of the hub shears and moments.

For noise reduction studies, the acoustic environment in the vicinity of the helicopter is characterized bythe noise decibel levels computed on a carpet plane located 1.15R beneath the rotor, as depicted in Fig. 4.Noise measured by a microphone placed on the right landing skid at the rear is used as the feedback signal

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to the controller. The sharp trailing edge con�guration, shown in Fig. 2, was chosen for the micro ap. Themicro ap, 1.5%c in height, slides in and out of a cavity, located at 6%c from the sharp trailing edge of theairfoil. Two di�erent con�gurations of micro aps on the rotor blade are considered in this study. The �rst

Table 1. Rotor parameters used for the computations.

Dimensional Rotor Data

R = 4:91 m

Mb = 27:35 kg

= 425 RPM

Nondimensional Main Rotor Data

Nb = 4 c = 0:05498R

Lb = 1:0 e = 0

XA = 0 �p = 2:5�

XIb = 0 XIIb = 0

IMB2 = 0 IMB3 = 0:0004

EI�� = 0:0302 EI�� = 0:0105

GJb = 0:0015 !F1 = 1:124

!F2 = 3:404 !F3 = 7:606

!L1 = 0:732 !L2 = 4:458

!T1 = 3:170 !T2 = 9:079

= 5:5 � = 0:07

�tw = �8�

Nondimensional Tail Rotor Data

Xt = 1:20 Zt = 0

Helicopter Data

CW = 0.005 fCdf = 0.031

XFA = 0:0 ZFA = 0:3

XFC = 0:0 ZFC = 0:3

con�guration, shown in Figure 5(a), has a single micro ap with 0.12R spanwise length centered at 0.75R. Thesecond con�guration, shown in Figure 5(b), has two micro aps each with 0.06R spanwise length centered at0.72R and 0.92R, respectively. Active control studies were also conducted using a 20%c conventional plain ap, shown in Fig. 1. The spanwise ap con�gurations considered are shown in Fig. 6.

V.A. Low-speed results

Various saturation approaches described earlier are compared in terms of their vibration, noise, and alsosimultaneous vibration and noise reduction capabilities at a heavy BVI descending ight condition withadvance ratio � = 0:15 and descent angle �D = 6:5�. Vibration reduction results are presented �rst.The non-dimensional 4/rev vibratory hub shears and moments obtained during active vibration reductionusing the four di�erent saturation approaches for the single plain ap con�guration are compared to thebaseline (no active control) vibration levels in Fig. 7. The TR approach yields only a 5% reduction in thevibration objective while causing some of the vibratory hub loads to increase from the baseline levels. TheSC approach yields a 38% reduction in the vibration objective. The AW and OPT approaches yield similarperformance yielding 76% and 78% reduction in the vibration objective, respectively. However, the OPTapproach takes only 10 control updates (80 rotor revolutions) to converge compared to over 100 controlupdates (800 rotor revolutions) taken by the AW approach. The ap de ection histories over one completerotor revolution corresponding to the various saturation approaches are shown in Fig. 8. The ap de ectionhistories corresponding to the AW and OPT approaches show similarities in the azimuthal locations of thepeaks and troughs. However, the maximum ap de ection obtained using the OPT approach is closer to

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R

1.15R

Y/R

X/R

-1

0

1

210-1-2

X

Y

Onboard Microphones

Carpet Plane

Retreating Side

SKID TIP

Advancing SideTop View

SKID MIDDLE

SKID REARBOOM

Figure 4. Microphone locations on and around the helicopter for noise measurements.

0.69R

0.12R

(a) Single Micro ap

0.69R

0.06R 0.06R0.14R

(b) Dual Micro ap

Figure 5. Single and dual spanwise con�gurations of the micro ap on the rotor blade

0.69R

0.12R

(a) Single plain ap

0.69R

0.06R 0.06R0.14R

(b) Dual plain ap

Figure 6. Single and dual spanwise con�gurations of the 20%c plain ap on the rotor blade

11 of 27


the saturation limit thus utilizing the ap to the maximum possible degree. This results from the fact thata small tolerance, typically of the order of 10�6, is used on the constraint violation in the OPT approach.

Baseline

Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

Non

-dim

ensi

onal

4/r

ev v

ibra

tory

hub

load

s

Long. shear Lat. shear RollingVert. shear Pitching Yawing

1 Plain flap, μ=0.15

Figure 7. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approaches forthe single plain ap con�guration at a heavy BVI descending ight condition with � = 0:15.

The 4/rev vibratory hub loads obtained during active vibration reduction using the dual plain apcon�guration are compared to the baseline levels in Fig. 9. The TR and SC approaches yield 81% and57% reduction in the vibration objective, respectively. The AW approach yields 95% reduction whereas theOPT approach yields the best performance with a 98% reduction in the vibration objective. Signi�cantlybetter performance obtained using the OPT approach is apparent in the vertical shear component. Theinboard and outboard ap de ection histories corresponding to the various saturation approaches are shownin Fig. 10. In the AW approach, the maximum de ection of the inboard ap is signi�cantly less than thesaturation limit. This under-utilization of one of the aps is primarily a result of the fact that the samecontrol weighting (cwu in Eq. (27)) is used for both the aps. Using a di�erent control weighting for thetwo micro aps results in a signi�cant increase in the computational time. By contrast, the two aps areoptimized individually in the OPT approach as is evident from the constraint inequalities in Eq. (29). Thus,the maximum de ection of both the aps is equal to the saturation limit within the tolerance limits. Thisfeature of the OPT approach facilitates the use of both the aps to the maximum possible extent resulting ina better vibration reduction performance.

Similar comparisons are also performed for the single and dual micro ap con�gurations. The 4/revvibratory hub loads obtained using the various saturation approaches for a single micro ap con�gurationare compared to the baseline levels in Fig. 11. The TR approach yields a 64% reduction in the vibrationobjective whereas the SC approach yields no signi�cant reduction. The AW and OPT approaches yieldsimilar performance with 71% and 70% reductions in the vibration objective, respectively. The micro apde ection histories corresponding to the various approaches are shown in Fig. 12 for one complete rotorrevolution. The ap de ection histories obtained from the SC, AW, and OPT approaches show similaritiesin the azimuthal locations of the peaks and troughs.

The 4/rev vibratory hub loads obtained using the various saturation approaches for the dual micro apcon�guration are compared in Fig. 13. The TR and SC approaches yield marginal performance with 13% and11% reductions in the vibration objective, respectively. The AW approach reduces the vibration objective by88% whereas the OPT approach yields the best performance with 97% reduction in the vibration objective.The inboard and outboard micro ap de ection histories corresponding to the various saturation approachesare shown in Fig. 14. As in the case of the plain aps, the AW approach under-utilizes the inboard controlsurface whereas the OPT approach utilizes both the control surfaces to the fullest possible extent resultingin a signi�cantly better vibration reduction performance.

The di�erent saturation approaches are also compared in terms of their BVI noise reduction capabilitiesusing the dual micro ap con�guration. Noise levels computed on the carpet plane (depicted in Fig. 4) duringactive noise reduction using the di�erent saturation approaches are compared to the baseline noise levels in

12 of 27


0 90 180 270 360

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−2

0

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eg]

(a) Truncation

0 90 180 270 360

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−2

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Azimuth [deg]

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(b) Scaling

0 90 180 270 360

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(c) Auto-weighting

0 90 180 270 360

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−2

0

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4

Azimuth [deg]

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p D

efle

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n [d

eg]

(d) Optimization

Figure 8. Single plain ap de ection histories corresponding to the various saturation approaches at a heavy BVIdescending ight condition with � = 0:15.

Baseline

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

Non

-dim

ensi

onal

4/r

ev v

ibra

tory

hub

load

s


Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

2 Plain flaps, μ=0.15

Figure 9. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approaches forthe dual plain ap con�guration at a heavy BVI descending ight condition with � = 0:15.

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0 90 180 270 360

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InboardOutboard

(a) Truncation

0 90 180 270 360

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Azimuth [deg]

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(b) Scaling

0 90 180 270 360

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(c) Auto-weighting

0 90 180 270 360

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4

Azimuth [deg]

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p D

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n [d

eg]

(d) Optimization

Figure 10. Dual plain ap de ection histories corresponding to the various saturation approaches at a heavy BVIdescending ight condition with � = 0:15.

Baseline

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

Non

-dim

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ev v

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tory

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load

s


Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

1 Microflap, μ=0.15

Figure 11. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the single micro ap con�guration at a heavy BVI descending ight condition with � = 0:15.

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0 90 180 270 360−0.5

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c]

(a) Truncation

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(b) Scaling

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n [%

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(c) Auto-weighting

0 90 180 270 360−0.5

0

0.5

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Azimuth [deg]

Mic

rofla

p D

efle

ctio

n [%

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(d) Optimization

Figure 12. Single micro ap de ection histories corresponding to the various saturation approaches at a heavy BVIdescending ight condition with � = 0:15.

Baseline

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

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-dim

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ev v

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tory

hub

load

s


Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

2 Microflaps, μ=0.15

Figure 13. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the dual micro ap con�guration at a heavy BVI descending ight condition with � = 0:15.

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0 90 180 270 360−0.5

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0.5

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n [%

c]

InboardOutboard

(a) Truncation

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Azimuth [deg]

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(b) Scaling

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Azimuth [deg]

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n [%

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(c) Auto-weighting

0 90 180 270 360−0.5

0

0.5

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Azimuth [deg]

Mic

rofla

p D

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(d) Optimization

Figure 14. Dual micro ap de ection histories corresponding to the various saturation approaches at a heavy BVIdescending ight condition with � = 0:15.

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Fig. 15. The TR and SC approaches yield similar performance with approximately 1 dB noise reduction onthe advancing side and up to 2 dB reduction on the retreating side of the rotor disk. By contrast, the AWand OPT approaches yield signi�cantly better performance with a 4-5 dB noise reduction on the advancingside and a 3-4 dB reduction on the retreating side of the rotor disk. The inboard and outboard micro apde ection histories corresponding to the various saturation approaches are shown in Fig. 16.

Noise Reduction, Truncation

Crossflow Position Y/R

−1 0 1

−1

−2

0

1

2

114

113

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116

114

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115

113112

Noise Reduction, Scaling

Noise Reduction, Auto-weighting

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110109

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112111

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( a. ) ( b. )

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120119118117116115114113112111110109108107106105104103102101100

B V I S PL - d B

Str

ea

mw

ise

Po

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Noise Reduction, Optimization

−1 0 1

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Baseline Simulation

Str

ea

mw

ise

Po

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−1 0 1

−1

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108

( d. )

113

112

−1

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1

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117

−1

−2

0

1

2


Figure 15. Reduction in noise levels obtained using the various saturation approaches for the dual micro ap con�gu-ration at a heavy BVI descending ight condition with � = 0:15.

The di�erent saturation approaches are also compared in terms of their simultaneous BVI noise andvibration reduction capabilities using the dual micro ap con�guration. For the simulations performed usingthe dual micro ap con�guration, the relative weighting factor W� = 0:6 is chosen as it yields an optimalreduction in the vertical hub shear and the feedback microphone noise levels. The noise levels computedon the carpet plane during simultaneous reduction using the di�erent saturation approaches are comparedto the baseline noise levels in Fig. 17. The TR approach reduces the noise levels by 1 dB on both theadvancing and retreating sides of the rotor disk. The SC, AW, and the OPT approaches yield similarperformance with 2 dB noise reduction on both the advancing and retreating sides. The 4/rev vibratoryhub loads obtained during simultaneous reduction using the various saturation approaches are compared tothe baseline levels in Fig. 18. The TR and SC approaches result in a 29% and 9% increase in the vibrationobjective, respectively. The AW approach yields 23% reduction whereas the OPT approach yields a 29%reduction in the vibration objective. Signi�cantly better performance obtained using the AW and OPTapproaches is apparent in the vertical hub shear component. The TR and SC approaches reduce the vertical

17 of 27


0 90 180 270 360−0.5

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rofla

p D

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n [%

c]

InboardOutboard

(a) Truncation

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Azimuth [deg]

Mic

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(b) Scaling

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Azimuth [deg]

Mic

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n [%

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(c) Auto-weighting

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Azimuth [deg]

Mic

rofla

p D

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n [%

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(d) Optimization

Figure 16. Dual micro ap de ection histories corresponding to the various saturation approaches during active noisereduction at a heavy BVI descending ight condition with � = 0:15.

18 of 27


hub shear by 3% and 10%, respectively, whereas, the AW and OPT approaches reduce the vertical hub shearby 34% and 37%, respectively. Overall, the AW and OPT approaches yield similar simultaneous BVI noiseand vibration reduction performance. However, the OPT approach requires signi�cantly less computationaltime to converge taking only 10 control updates (80 rotor revolutions) compared to over 100 control updates(800 rotor revolutions) taken by the AW approach. The inboard and outboard micro ap de ection historiescorresponding to the various saturation approaches are shown in Fig. 19. In the AW approach, the maximumde ection of the inboard micro ap is less than the saturation limit.

Simultaneous Reduction, Truncation


−1 0 1

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113112

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( a. ) ( b. )

( c. )

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120119118117116115114113112111110109108107106105104103102101100

B V I S PL - d B

Str

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ise

Po

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−1

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Simultaneous Reduction, Optimization

−1 0 1

−1 0 1

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Baseline Simulation

Str

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ise

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−1

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112

113

112

115

115

113

Figure 17. Reduction in noise levels obtained during simultaneous BVI noise and vibration reduction using the varioussaturation approaches for the dual micro ap con�guration at a heavy BVI descending ight condition with � = 0:15.

V.B. High-speed Results

In this section, the vibration reduction performance of the various saturation approaches is compared at ahigh speed level ight condition with � = 0:3. The 4/rev vibratory hub loads obtained using the varioussaturation approaches for the single plain ap con�guration are compared to the baseline levels in Fig. 20. TheTR and SC approaches yield 82% and 71% reduction in the vibration objective, respectively. Comparatively,the AW and OPT approaches yield signi�cantly better performance reducing the vibration objective by 93%and 97%, respectively. The de ection time histories corresponding to the various saturation approaches overone rotor revolution are shown in Fig. 21. A relatively good performance obtained from the TR approachcan be attributed to the fact that only a small portion of the ap de ection shown in Fig. 21(a) is being

19 of 27


Baseline

0

0.0005

0.001

0.0015

0.002

0.0025

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Non

-dim

ensi

onal

4/r

ev v

ibra

tory

hub

load

s


Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

2 Microflaps, Simultaneous Redu�on,μ=0.15

Figure 18. Reduction in 4/rev vibratory hub shears and moments obtained during simultaneous BVI noise and vibrationreduction using the various saturation approaches for the dual micro ap con�guration at a heavy BVI descending ightcondition with � = 0:15.

0 90 180 270 360−0.5

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0.5

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Azimuth [deg]

Mic

rofla

p D

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n [%

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InboardOutboard

(a) Truncation

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Azimuth [deg]

Mic

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(b) Scaling

0 90 180 270 360−0.5

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Mic

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(c) Auto-weighting

0 90 180 270 360−0.5

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Azimuth [deg]

Mic

rofla

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(d) Optimization

Figure 19. Dual micro ap de ection histories corresponding to the various saturation approaches at a heavy BVIdescending ight condition with � = 0:15.

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truncated. The ap de ection histories corresponding to the SC, AW, and OPT approaches show a qualitativeresemblance with similar azimuthal locations for the peaks and troughs.

Baseline

0

0.0005

0.001

0.0015

0.002

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0.003

Non

-dim

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onal

4/r

ev v

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tory

hub

load

s


Trunca�on

Scaling

Auto-weigh�ng

Op�miza�on

1 Plain flap, μ=0.30

Figure 20. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the single plain ap con�guration at a high-speed ight condition with � = 0:3.

Vibratory hub loads obtained using the various saturation approaches for the dual plain ap con�gurationare shown in Fig. 22. The TR approach causes an increase in the longitudinal and lateral shears and yieldsno signi�cant reduction in the vibration objective. The SC approach reduces the vibration objective by 62%whereas the AW and the OPT approaches yield exceptional performance with 95% and 99% reductions inthe vibration objective, respectively. The inboard and outboard ap de ection time histories correspondingto the various approaches are shown in Fig. 23. In the TR approach, signi�cant portions of the inboardand outboard ap de ections are truncated, as shown in Fig. 23(a), resulting in its poor performance. Theoutboard ap is signi�cantly under-utilized by the AW approach whereas the OPT approach utilizes boththe aps to the maximum possible extent.

Similar comparisons are also performed using single and dual micro ap con�gurations. The 4/rev vi-bratory hub loads for the various saturation approaches are compared in Fig. 24 for the single micro apcon�guration. The TR approach reduces the vibration objective by 38%. The SC approach causes a sig-ni�cant increase in the vertical hub shear resulting in a 5% increase in the vibration objective. The AWand OPT approaches yield 91% and 94% reductions in the vibration objective, respectively. The micro apde ection time histories corresponding to the various saturation approaches are shown in Fig. 25 for onecomplete rotor revolution. The micro ap de ection histories corresponding to the AW and OPT approachesexhibit similarity in the overall shape.

Vibratory hub loads obtained from the di�erent saturation approaches for the dual micro ap con�gurationare shown in Fig. 26. The TR and SC approaches yield 25% and 28% reductions in the vibration objective,respectively. However, both of them cause a small increase in the vertical hub shear. The AW and theOPT approaches yield exceptional performance with 94% and 98% reductions in the vibration objective,respectively. The micro ap de ection histories corresponding to the various saturation approaches are shownin Fig. 27. The AW approach signi�cantly under-utilizes the outboard micro ap whereas the OPT approachutilizes both micro aps to the maximum possible extent.

VI. Summary and Conclusions

The e�ect of actuator saturation on the active vibration and noise reduction performance of the con-ventional trailing-edge plain aps and the micro aps is examined. A new approach to handling actuatorsaturation in the HHC algorithm based on constrained nonlinear optimization techniques is developed. Vi-bration reduction performance of this approach is compared to the existing approaches at various ightconditions. Noise reduction and simultaneous vibration and noise reduction performance of the various

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Figure 21. Single plain ap de ection histories corresponding to the various saturation approaches at a high-speed ight condition with � = 0:3.

Baseline

0

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0.001

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Trunca�on

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Auto-weigh�ng

Op�miza�on

2 Plain flaps, μ=0.30

Figure 22. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the dual plain ap con�guration at a high-speed ight condition with � = 0:3.

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0 90 180 270 360

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(c) Auto-weighting

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Figure 23. Dual plain ap de ection histories corresponding to the various saturation approaches at a high-speed ightcondition with � = 0:3.

Baseline

Non

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1 Microflap, μ=0.30

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Figure 24. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the single micro ap con�guration at a high-speed ight condition with � = 0:3.

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Figure 25. Single micro ap de ection histories corresponding to the various saturation approaches at a high-speed ight condition with � = 0:3.

Baseline

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2 Microflaps, μ=0.30

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

Figure 26. Reduction in 4/rev vibratory hub shears and moments obtained using the various saturation approachesfor the dual micro ap con�guration at a high-speed ight condition with � = 0:3.

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Figure 27. Dual micro ap de ection histories corresponding to the various saturation approaches at a high-speed ightcondition with � = 0:3.

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approaches is compared at a heavy BVI low-speed descending ight condition. The principal �ndings andconclusions of the study are summarized below.

1. The truncation and scaling approaches are inconsistent and yield only limited vibration or noise reduc-tion using plain trailing-edge aps or micro aps. By comparison, the auto-weighting and optimizationapproaches show very good performance.

2. In the case of multiple control surfaces, the optimization approach utilizes all of them to the maximumpossible extent producing signi�cantly better vibration reduction performance. In the auto-weightingapproach, the weighting on the control inputs corresponding to the various control surfaces is identical.This results in an optimal control input only for one of the control surfaces leaving the others under-utilized. In the optimization approach, the de ections corresponding to the various control surfaces areoptimized independently and hence are utilized to the maximum possible extent resulting in a bettervibration reduction performance.

3. The optimization approach takes signi�cantly less computational time compared to the auto-weightingapproach. This is primarily because the auto-weighting approach relies on an a priori guess of theupper bound of the optimal control weighting and involves iteratively adjusting the weighting matrixR until a control input satisfying the saturation constraints is obtained.

Thus the optimization based approach is the best option for handling actuator saturation constraints inthe HHC algorithm during on-blade vibration and noise reduction using aps and micro aps, especially forthe case of multiple control surfaces.

Acknowledgments

This research was supported by the Vertical Lift Research Center of Excellence (VLRCOE) sponsoredby NRTC and U.S. Army with Dr. M. Rutkowski as grant monitor.

References

1Friedmann, P. P. and Millott, T. A., \Vibration Reduction in Rotorcraft Using Active Control: A Comparison of VariousApproaches," Journal of Guidance, Control, and Dynamics, Vol. 18, No. 4, July-August 1995, pp. 664{673.

2Swanson, S. M., Jacklin, S. A., Blaas, A., Niesl, G., and Kube, R., \Reduction of Helicopter BVI Noise, Vibration, andPower Consumption through Individual Blade Control," Proceedings of the 51st Annual Forum of the American HelicopterSociety, Fort Worth, TX, May 1995, pp. 662{680.

3Jacklin, S. A., Haber, A., de Simone, G., Norman, T., Kitaplioglu, C., and Shinoda, P., \Full-Scale Wind Tunnel Test ofan Individual Blade Control System for a UH-60 Helicopter," Proceedings of the 58th Annual Forum of the American HelicopterSociety, Montreal, Canada, June 2002.

4Millott, T. A. and Friedmann, P. P., Vibration Reduction in Helicopter Rotors Using an Actively Controlled Partial SpanTrailing Edge Flap Located on the Blade, NASA CR 4611, June 1994.

5Koratkar, N. A. and Chopra, I., \Wind Tunnel Testing of a Smart Rotor Model with Trailing Edge Flaps," Journal ofthe American Helicopter Society, Vol. 47, No. 4, Oct. 2002, pp. 263{272.

6Patt, D., Liu, L., and Friedmann, P. P., \Simultaneous Vibration and Noise Reduction in Rotorcraft Using AeroelasticSimulation," Journal of the American Helicopter Society, Vol. 51, No. 2, April 2006, pp. 127{140.

7Straub, F. K., Anand, V., Birchette, T. S., and Lau, B. H., \Wind Tunnel Test of the SMART Active Flap Rotor,"Proceedings of the 65th American Helicopter Society Annual Forum, Grapevine,TX, May 2009.

8Shin, S. J., Cesnik, C. E. S., and Hall, S. R., \Closed-Loop Control Tests of the NASA/ARMY/MIT Active Twist Rotorfor Vibration Reduction," Proceedings of the American Helicopter Society 59th Annual Forum, Phoenix, AZ, June 2003.

9Bernhard, A. and Chopra, I., \Hover Test of Mach-Scale Active Twist Rotor Using Piezo-Bending-Torsion Actuators,"Journal of Aircraft , Vol. 39, No. 4, 2002, pp. 678{688.

10Liu, L., Padthe, A. K., and Friedmann, P. P., \Computational Study of Micro aps with Application to VibrationReduction in Helicopter Rotors," AIAA Journal , Vol. 49, No. 7, July 2011, pp. 1450{1465.

11Padthe, A. K., Liu, L., and Friedmann, P. P., \Numerical Evaluation of Micro aps for On Blade Control of Noise and Vi-bration," Proceedings of the 52nd AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and Materials Conference,Denver, CO, April 2011.

12Min, B. Y., Sankar, L., and Bauchau, O. A., \A CFD-CSD Coupled Analysis of Hart II Rotor Vibration Reduction UsingGurney Flaps," Proceedings of the 66th American Helicopter Society Annual Forum, Phoenix, AZ, May 11-13 2010.

13Bernstein, D. S. and Michel, A. N., \A Chronological Bibliography on Saturating Actuators," International Journal ofRobust and Nonlinear Control , Vol. 5, 1995, pp. 375{380.

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14Myrtle, T. F. and Friedmann, P. P., \Application of a New Compressible Time Domain Aerodynamic Model to VibrationReduction in Helicopters Using an Actively Controlled Flap," Journal of the American Helicopter Society, Vol. 46, No. 1, Jan.2001, pp. 32{43.

15Patt, D., Liu, L., Chandrasekar, J., Bernstein, D. S., and Friedmann, P. P., \Higher-Harmonic-Control Algorithm forHelicopter Vibration Reduction Revisited," Journal of Guidance, Control, and Dynamics, Vol. 28, No. 5, September-October2005, pp. 918{930.

16Johnson, W., Self-Tuning Regulators for Multicyclic Control of Helicopter Vibrations, NASA Technical Paper 1996, 1982.17Cribbs, R. and Friedmann, P. P., \Actuator Saturation and Its in uence on Vibration Reduction by Actively Controlled

Flaps," AIAA Paper No. 2001-1467. Proceedings of the 42nd AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamicsand Materials Conference, Seattle, WA, April 2001.

18Roget, B. and Chopra, I., \Closed-Loop Test of a Rotor with Individually Controlled Trailing-Edge Flaps for VibrationReduction," Journal of the American Helicopter Society, Vol. 55, No. 1, 2010.

19Glaz, B., Friedmann, P. P., Liu, L., Kumar, D., and Cesnik, C. E. S., \The AVINOR Aeroelastic Simulation Code and ItsApplication to Reduced Vibration Composite Rotor Blade Design," Proceedings of the 50th AIAA/ASME/ASCE/AHS/ACSStructures, Structural Dynamics and Materials Conference, Palm Springs, CA, May 2009, AIAA Paper No. 2009-2601.

20de Terlizzi, M. and Friedmann, P. P., \Active Control of BVI Induced Vibrations Using a Re�ned Aerodynamic Modeland Experimental Correlation," American Helicopter Society 55th Annual Forum Proceedings, Montreal, Canada, May 25-271999, pp. 599{615.

21Liu, L., Padthe, A., Friedmann, P. P., Quon, E., and Smith, M., \Unsteady Aerodynamics of an Airfoil/Flap Combinationon a Helicopter Rotor Using CFD and Approximate Methods," Journal of American Helicopter Society, Vol. 56, No. 3, July2011, pp. 1{13.

22Brenter, K., A Computer Program Incorporating Realistic Blade Motions and Advanced Acoustic Formulation, NASATechnical Memorandum, Vol. 87721 1986.

23Patt, D., Liu, L., and Friedmann, P. P., \Rotorcraft Vibration Reduction and Noise Prediction Using a Uni�ed AeroelasticResponse Simulation," Journal of the American Helicopter Society, Vol. 50, No. 1, Jan. 2005, pp. 95{106.

24Abbott, I. H., Doenho�, A. E. V., and Stivers, L. S., Summary of Airfoil Data, NACA Report 824, 1945.25Fletcher, R., Practical Methods of Optimization, John Wiley and Sons, 1987.26Schittkowski, K., \NLQPL: A FORTRAN-Subroutine Solving Constrained Nonlinear Programming Problems," Annals

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